Planar Riemann surface

Elementary properties

A closed 1-form ω is exact if and only if ∫_γ ω = 0 for every closed Jordan curve γ.

A closed Jordan curve γ on a Riemann surface separates the surface into two disjoint connected regions if and only if ∫_γ ω = 0 for every closed 1-form ω of compact support.

A Riemann surface is planar if and only if every closed 1-form of compact support is exact.

Every connected open subset of a planar Riemann surface is planar.

Every simply connected Riemann surface is planar.

Uniformization theorem

Koebe's Theorem. A compact planar Riemann surface X is conformally equivalent to the Riemann sphere. A non-compact planar Riemann surface X is conformally equivalent to the complex plane with isolated points or closed intervals parallel to the real axis removed.

The harmonic function U. If X is a Riemann surface and P is a point on X with local coordinate z, there is a unique real-valued harmonic function U on X {P} such that U(z) – Re z⁻¹ is harmonic near z = 0 (the point P) such that dU is square integrable on the complement of a neighbourhood of P. Moreover if h is any real-valued smooth function on X vanishing in a neighbourhood of the pole of U with ||dh||² = ∫_X dh∧∗dh < ∞, then (dU,dh) = ∫_X dU ∧ *dh = 0.

The conjugate harmonic function V.

The meromorphic function f.

Koebe's separation argument. Let φ and ψ be smooth bounded real-valued functions on R with bounded first derivatives such that φ'(t) > 0 for all t ≠ 0 and φ vanishes to infinite order at t = 0 while ψ(t) > 0 for t in (a,b) while ψ(t) ≡ 0 for t outside (a,b) (here a = −∞ and b = +∞ are allowed). Let X be a Riemann surface and W an open connected subset with a holomorphic function g = u + iv differing from f by a constant such that g(W) lies in the strip a < Im z < b. Define a real-valued function by h = φ(u)ψ(v) on W and 0 off W. Then h, so defined, cannot be a smooth function; for if so

where M = sup (|φ|, |φ'|, |ψ|, |ψ'|), and contradicting the orthogonality condition on U.

Connectivity and level curves. The level curves for V divide X into two open connected regions. The open set between two level curves of V is connected. The level curves for U and V through any regular point of f divide X into four open connected regions, each containing the regular point and the pole of f in their closures.

Univalence of f at regular points. The function f takes different values at distinct regular points (where df ≠ 0).

Regularity of f. The meromorphic function f is regular at every point except the pole.

The complement of the image of f. Either the image of f is the whole Riemann sphere C ∪ ∞, in which case the Riemann surface is compact and f gives a conformal equivalence with the Riemann sphere; or the complement of the image is a union of closed intervals and isolated points, in which case the Riemann surface is conformally equivalent to a horizontal slit region.

Applications

Koebe's uniformization theorem for planar Riemann surfaces implies the uniformization theorem for simply connected Riemann surface. Indeed, the slit domain is either the whole Riemann sphere; or the Riemann sphere less a point, so the complex plane after applying a Möbius transformation to move the point to infinity; or the Riemann sphere less a closed interval parallel to the real axis. After applying a Möbius transformation, the closed interval can be mapped to [–1,1]. It is therefore conformally equivalent to the unit disk, since the conformal mapping g(z) = (z + z⁻¹)/2 maps the unit disk onto C  [−1,1].

Koebe's theorem also implies that any finitely connected bounded region in the plane is conformally equivalent to the open unit disk with finitely many smaller disjoint closed disks removed, or equivalently the extended complex plane with finitely many disjoint closed disks removed. This result is known as Koebe's "Kreisnormierungs" theorem.

Following Goluzin (1969) it can be deduced from the parallel slit theorem using a variant of Carathéodory's kernel theorem and Brouwer's theorem on invariance of domain. Goluzin's method is a simplification of Koebe's original argument.

In fact any conformal mapping of such a circular domain onto another circular domain is necessarily given by a Möbius transformation. To see this, it can be assumed that both domains contain the point ∞ and that the conformal mapping f carries ∞ onto ∞. The mapping functions can be continued continuously to the boundary circles. Successive inversions in these boundary circles generate Schottky groups. The union of the domains under the action of both Schottky groups define dense open subsets of the Riemann sphere. By the Schwarz reflection principle. f can be extended to a conformal map between these open dense sets. Their complements are the limit sets of the Schottky groups. They are compact and have measure zero. The Koebe distortion theorem can then be used to prove that f extends continuously to a conformal map of the Riemann sphere onto itself. Consequently, f is given by a Möbius transformation.

Now the space of circular domains with n circles has dimension 3n – 2 (fixing a point on one circle) as does the space of parallel slit domains with n parallel slits (fixing an endpoint point on a slit). Both spaces are path connected. The parallel slit theorem gives a map from one space to the other. It is one-one because conformal maps between circular domains are given by Möbius transformations. It is continuous by the convergence theorem for kernels. By invariance of domain, the map carries open sets onto open sets. The convergence theorem for kernels can be applied to the inverse of the map: it proves that if a sequence of slit domains is realisable by circular domains and the slit domains tend to a slit domain, then the corresponding sequence of circular domains converges to a circular domain; moreover the associated conformal mappings also converge. So the map must be onto by path connectedness of the target space.

An account of Koebe's original proof of uniformization by circular domains can be found in Bieberbach (1953). Uniformization can also be proved using the Beltrami equation. Schiffer & Hawley (1962) constructed the conformal mapping to a circular domain by minimizing a nonlinear functional—a method that generalized the Dirichlet principle.

Koebe also described two iterative schemes for constructing the conformal mapping onto a circular domain; these are described in Gaier (1964) and Henrici (1986). In fact suppose a region on the Riemann sphere is given by the exterior of n disjoint Jordan curves and that ∞ is an exterior point. Let f₁ be the Riemann mapping sending the outside of the first curve onto the outside of the unit disk, fixing ∞. The Jordan curves are transformed by f₁ to n new curves. Now do the same for the second curve to get f₂ with another new set of n curves. Continue in this way until f_n has been defined. Then restart the process on the first of the new curves and continue. The curves gradually tend to fixed circles and for large N the map f_N approaches the identity; and the compositions f_N ∘ f_N−1 ∘ ⋅⋅⋅ ∘ f₂ ∘ f₁ tend uniformly on compacta to the uniformizing map.

Uniformization by parallel slit domains holds for arbitrary connected open domains in C; Koebe (1908) conjectured (Koebe's "Kreisnormierungsproblem") that a similar statement was true for uniformization by circular domains. He & Schramm (1993) proved Koebe's conjecture when the number of boundary components is countable; although proved for wide classes of domains, the conjecture remains open when the number of boundary components is uncountable. Koebe (1936) also considered the limiting case of osculating or tangential circles which has continued to be actively studied in the theory of circle packing.